Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-5x-4y &= 7 \\ -2x-4y &= 1\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $1$ $\begin{align*}5x+4y &= -7\\ -2x-4y &= 1\end{align*}$ Add the top and bottom equations. $3x = -6$ Divide both sides by $3$ and reduce as necessary. $x = -2$ Substitute $-2$ for $x$ in the top equation. $-5( -2)-4y = 7$ $10-4y = 7$ $-4y = -3$ $y = \dfrac{3}{4}$ The solution is $\enspace x = -2, \enspace y = \dfrac{3}{4}$.